Section 91
Definitions: Relations and Functions
A
relation
is simply a set of
ordered pairs
. Usually, we talk about relations on sets of numbers, but not always.
Easy Example:
You could have a relation between the set of all names and the set of whole numbers. A name N is related to a number
x
if and only if N has fewer than
x
letters.
So, (Raj, 5) is in the relation, but (Abdullah, 7) is not.
Easy Example:
Here is a relation on the set of real numbers. Suppose
x
is related to
y
if
x
<
y
,
and not related otherwise.
The following table shows some ordered pairs which are in the relation, and some which are not.
Related

Not Related

(1, 6)

(3, –2)

(5, 5.001)

( –8, –7)

( –1, 9999)

(4, 4)

INPUTOUTPUT TABLES
One way in which relations are commonly displayed is in an inputoutput table. The idea is, you input some number
x
, and you get out some
y
.
Input

Output

0

0

1

3

2

6

3

9

10

30

–5

–15

This table describes a relation containing the ordered pairs (0, 0), (1, 3), (2, 6), (3, 9), (10, 30), (–5, –15).
If the same input always gives the same output, then the relation is called a
function
. Otherwise it is not a function. The relation in the table above is a function (it is okay if two different inputs give the same output). The relation in the table below is not a function because the same input 1 gives the output 5 the first time and 0 the second time.
Input

Output

0

0

1

5

2

0

3

15

1

0

–5

–15

THE VERTICAL LINE TEST
If you have a graph of a relation, you can use the vertical line test to decide whether or not it is a function. This means: if there is any
vertical line
which intersects the graph in more than one point, then the relation is not a function.

The above relation (in blue) represents a function: every vertical line intersects it only once.


The above relation (in blue) is not a function: there are many vertical lines that intersect it more than once.

The
domain
is the set of all input values that make "sense"; the
range
is the set of all possible output values.
FUNCTION NOTATION
We can call the input
x
, the rule
f
, and then the output is
f
(
x
)
.
This DOES NOT mean "
f
times
x
" , it's just a notation device to record the input and output. For example, if
f
(
x
) =
x
^{
2
}
, then
f
(3) = 3
^{
2
}
= 9, not
f
times 3 (meaningless).
Think of
f
(
x
) =
x
^{
2
}
as
f
( ) = ( )
^{
2
}
; that way you can safely plug in negative numbers or even other expressions. For example:
f
(
–
5) = (
–
5)
^{
2
}
= 25
f
(
x
+
h
) = (
x
+
h
)
^{
2
}